- Email: [email protected]
A test for a conjunction
Statistics & Probability Letters 47 (2000) 135 – 140
A test for a conjunction ( K.J. Worsley a;∗ , K.J. Fristonb a Department
b Wellcome
of Mathematics and Statistics, McGill University, Montreal, Canada H3A 2K6 Department of Cognitive Neurology, Institute of Neurology, Queen Square, London WC1N 3BG, UK Received March 1999
Abstract A conjunction is de ned in the brain mapping literature as the occurrence of the same event at the same location in two or more independent 3D brain images. The images are smooth isotropic 3D random elds of test statistics, and the event occurs when the image exceeds a xed high threshold. We give a simple approximation to the probability of a conjunction occurring anywhere in a xed region, so that we can test for a local increase in the mean of the images at the same unknown location in all images, a generalization of the split-t test. This is the corollary to a more general result c 2000 Elsevier Science B.V. on the expected Minkowski functionals of the set of points where a conjunction occurs. All rights reserved MSC: 60G60; 62M09 Keywords: Split-t test; Random eld; Euler characteristic; Integral geometry; Stereology
1. Introduction Let Xi (t) be the value of image i at location t ∈ R D ; 16i6n, and let x be a xed threshold. The set of points where a conjunction occurs is C = {t ∈ S: Xi (t)¿x for all 16i6n}: An example is shown in Fig. 1 for n = 6. We are interested in the probability that C is not empty, that is, the probability that all images exceed the threshold at some point inside S, or that the maximum over t ∈ S
( Research supported by the Natural Sciences and Engineering Research Council of Canada, Fonds pour la Formation des Chercheurs et 1’Aide a la Recherche de Quebec, and the Wellcome Trust. ∗
Corresponding author. Tel.: +1-514-398-3842; fax: +1-514-398-3899. E-mail address: [email protected] (K.J. Worsley)
c 2000 Elsevier Science B.V. All rights reserved 0167-7152/00/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 9 9 ) 0 0 1 4 9 - 2
136
K.J. Worsley, K.J. Friston / Statistics & Probability Letters 47 (2000) 135 – 140
Fig. 1. Conjunction of n = 6 f MRI images during a visual task (only one slice of the 3D data is shown). The excursion sets of each Xi (t)¿1:64; i = 1; : : : ; 6 are shown in white on a background of brain anatomy (top). The set of conjunctions C is the intersection of these sets (bottom). The visual cortex at the back of the brain appears in C, but the most interesting feature is the appearance of the lateral geniculate nuclei (LGN) (arrows).
of the minimum over i of Xi (t) exceeds x: P{C 6= ∅} = P max min Xi (t)¿x : t ∈ S 16i6n
(1)
If the images are independent stationary random elds then the expected Lebesgue measure or volume of C is E{|C|}=pn |S|;
(2)
where p = P{Xi (t)¿x}. Our main result is that (2) holds if Lebesgue measure is replaced by a vector of Minkowski functionals, and p is replaced by a matrix of Euler characteristic intensity functions for the random eld. This gives (2) as a special case, and other interesting quantities such as the expected surface area of C, which comes from the (D − 1)-dimensional Minkowski functional. But the component of most interest to us is the zero-dimensional Minkowski functional, or Euler characteristic (EC). For high thresholds, the expected EC of C is a very accurate approximation to probability (1) that we seek (Adler, 1999). We apply this result to some real data in brain mapping in Section 5. This also allows us to set the level of the split-t test. Shaywitz et al. (1995) used this test to determine whether the functional organization of the brain for language di ered according to sex. Thirty eight independent
K.J. Worsley, K.J. Friston / Statistics & Probability Letters 47 (2000) 135 – 140
137
f MRI images were randomly divided into n = 2 groups, and an image Xi (t) of t-statistics was calculated for each group i =1; 2. The split-t test rejects if the null hypothesis is rejected for both groups at the same point t, that is, if X1 (t)¿x and X2 (t)¿x for some threshold x, taken as the upper level 5% point of the t-distribution with 17 degrees of freedom. The resulting image of conjunctions C appears on the cover of Nature that contains Shaywitz et al. (1995). 2. Integral geometry and stereology In this section we shall state some results from integral geometry and stereology that will be used to prove our main result (see, for example, Santalo, 1976). Let i (A) be the ith Minkowski functional of a set A ⊂ R D , scaled so that it is invariant under embedding of A into any higher-dimensional Euclidean space. If A has a twice di erentiable boundary @A, then it can be de ned as follows. Let si = 2i=2 = (i=2) be the surface area of a unit (i − 1)-sphere in R i . For M an m × m matrix let detr j (M ) denote the sum of the determinant of all j × j principal minors of M , so that detr m (M ) = det (M ); detr 1 (M ) = tr(M ) and we de ne detr 0 (M ) = 1. Let Q be the (D − 1) × (D − 1) curvature matrix of @A. Then for 06i6D, Z 1 i (A) = detr D−1−i (Q) dt sD−i @A and de ne D (A) = |A|. Note that 0 (A) is the EC of A by the Gauss–Bonnet Theorem, and D−1 (A) is half the surface area of A. For example, the Minkowski functionals of a ball A of radius r in R 3 are 0 (A) = 1;
1 (A) = 4r;
2 (A) = 2r 2 ;
We shall use the result that any set functional
3 (A) = (4=3)r 3 :
(3)
(A) that obeys the additivity rule
(A ∪ B) = (A) + (B) − (A ∩ B)
(4)
is a linear combination of the Minkowski functionals. Let A; B ⊂ R D , then the Kinematic Fundamental Formula of integral geometry relates the integrated EC of the intersection of A and B to their Minkowski functionals: Z D X i (A)D−i (B) 0 (A ∩ B) = s2 : : : sD ; (5) ciD i=0 where the integral is over all rotations and translations of A, keeping B xed, and ciD =
(1=2) ((D + 1)=2) : ((i + 1)=2) ((D − i + 1)=2)
3. Random elds If X (t); t ∈ R D , is an isotropic random eld with excursion set A = {t: X (t)¿x} then E{0 (A ∩ S)} =
D X
i i (S)
(6)
i=0
for some constant i . This follows from the fact that (S) = E{0 (S ∩ A)} obeys the additivity rule (4), since 0 does, so it must be a linear combination of the Minkowski functionals i (S). The coecients i , called Euler characteristic (EC) intensities in R i , can be evaluated for a variety of random elds (Adler, 1981; Worsley, 1994, 1999; Siegmund and Worsley, 1995; Cao and Worsley, 1999a,b). For example, for a
138
K.J. Worsley, K.J. Friston / Statistics & Probability Letters 47 (2000) 135 – 140
Gaussian random eld with E{X (t)} = 0; Var{X (t)} = 0; Var{@X (t)=@t} = I , where I is the D×D identity matrix, then 0 = P{X (t)¿x} and for i ¿ 0 2
i = i=2 (2)−(i+1)=2 Hei−1 (x)e−x =2 ;
(7)
where Hej (x) is the Hermite polynomial of degree j in x. We now extend (6) to higher Minkowski functionals. Lemma 1. E{i (A ∩ S)} =
D X
cij j−i j (S):
j=i
Proof. Clearly,
(S) = E{i (A ∩ S)} obeys the additivity rule (4), since i does, so we can write
E{i (A ∩ S)} =
D X
aij j (S)
(8)
j=0
for some constants aij that do not depend on S. To evaluate these constants, replace S in (8) by Ek , a bounded convex set in a k-plane, k6D. Since the ith Minkowski functional of a set in R k is zero for i ¿ k, then aik is zero for i ¿ k. For i6k, D
E{i (A ∩ Ek )} X j (Ek ) = → aik aij k (Ek ) k (Ek ) j=i
(9)
as Ek → R k . We can thus interpet aik as the density of the ith Minkowski functional of the excursion set of X in R k . To evaluate aik , apply the Kinematic Fundamental Formula (5) to A ∩ ED and S: Z 0 ((A ∩ ED ) ∩ S) = s2 : : : sD
D X i (A ∩ ED )D−i (S) i=0
ciD
:
Dividing both sides by s2 : : : sD D (ED ) and taking limits, we get E{0 (A ∩ S)} =
D X i=0
lim
ED →R
D
E{i (A ∩ ED )} D−i (S) : D (ED ) ciD
Comparing this with (6) we get lim
ED →R
D
E{i (A ∩ ED )} = ciD D−i D (ED )
and combining this with (9), we get for i6j, aij = cij j−i : Substituting into (8) completes the proof.
K.J. Worsley, K.J. Friston / Statistics & Probability Letters 47 (2000) 135 – 140
139
4. Geometry of the set of conjunctions Theorem 2. Let bi = ((i + 1)=2)= (1=2) and de ne the upper triangular Toeplitz matrix Rk and the vector (B) by 0k =b0 1k =b1 · · · 0 (B)b0 Dk =bD 0 (B)b 0k =b0 · · · (D−1)k =bD−1 1 1 ; ; (B) = Rk = . . . . .. .. .. .. . .. 0 0 ··· 0k =b0 D (B)bD where ik is the EC intensity of Xk (t) in R i ; 16k6n, and B ⊂ R D . Then ! n Y Ri (S): E{(C)} = i=1
Proof. The proof follows by induction on n. From the lemma, we see that it is clearly true for n = 1. Let Ak be the excursion set for Xk (t), so that C = A1 ∩ · · · ∩ An ∩ S. If the result is true for n = k then by rst conditioning on Ak+1 and replacing S by Ak+1 ∩ S we get ! ! k k Y Y E{(A1 ∩ · · · ∩ Ak ∩ (Ak+1 ∩ S)} = Ri E{(Ak+1 ∩ S)} = Ri Rk+1 (S) i=1
i=1
by the result for n = 1. This completes the proof. Comparing this result with (2) we see that it has the same form, with volume replaced by the vector of weighted Minkowski functionals, and probability replaced by the matrix of weighted EC intensities. The last element is the same as in (2), and the rst element is the expected EC of the set of conjunctions that we shall use as an approximation to the probability of a conjunction anywhere in S: ! n Y (10) Ri (S) P{C 6= 0} = P max min Xi (t)¿x ≈ E{0 (C)} = (1; 0; : : : ; 0) t ∈ S 16i6n
i=1
for high thresholds x. 5. Application We shall apply the result to some D = 3 dimensional functional magnetic resonance imaging (f MRI) data fully described in Friston et al. (1999). The purpose of the experiment was to determine those regions of the brain that were consistently stimulated by all subjects while viewing a pattern of radially moving dots. To do this, subjects were presented with a pattern of moving dots, followed by a pattern of stationary dots, and this was repeated 10 times, during which a total of 120 3D f MRI images were obtained at the rate of one every 3.22 s. For each subject i and at every point t ∈ R 3 , a test statistic Xi (t) was calculated for comparing the f MRI response between the moving dots and the stationary dots. Under the null hypothesis of no di erence, Xi (t) was modeled as an isotropic Gaussian random eld with zero mean, unit variance and = 4:68 cm−2 . A threshold of x = 1:64, corresponding to an uncorrected level 5% test, was chosen, and the excursion sets for each subject are shown in Fig. 1, together with their intersection, which forms the set of conjunctions C. The search region S is the whole brain area that was scanned, which was an approximate spherical region with a volume of |S| = 1226 cm3 . Finally, the approximate probability of a conjunction, calculated from (3), (7)
140
K.J. Worsley, K.J. Friston / Statistics & Probability Letters 47 (2000) 135 – 140
and (10), is 0.0126. We can thus conclude, at the 1.26% level, that conjunctions have occured in the visual cortex, and more interesting, the lateral geniculate nuclei (see Fig. 1). References Adler, R.J., 1981. The Geometry of Random Fields. Wiley, New York. Adler, R.J., 1999. On excursion sets, tube formulae, and maxima of random elds. Ann. Appl. Probab., in press. Cao, J., Worsley, K.J., 1999a. The detection of local shape changes via the geometry of Hotelling’s T 2 elds. Ann. Statist., in press. Cao, J., Worsley, K.J., 1999b. The geometry of correlation elds with an application to functional connectivity of the brain. Ann. Appl. Probab., in press. Friston, K.J., Holmes, A.P., Price, C.J., Buchel, C., Worsley, K.J., 1999. Multi-subject f MRI studies and conjunction analyses. Neuroimage 10, 385–396. Santalo, L.A., 1976. Integral geometry and geometric probability. In: Rota, G.-C. (Ed.), Encyclopedia of Mathematics and its Applications, Vol. 1. Addison-Wesley, Reading, MA. Shaywitz, B.A., Shaywitz, S.E., Pugh, K.R., 1995. Sex di erences in the functional organization of the brain for language. Nature 373, 607–609. Siegmund, D.O., Worsley, K.J., 1995. Testing for a signal with unknown location and scale in a stationary Gaussian random eld. Ann. Statist. 23, 608–639. Worsley, K.J., 1994. Local maxima and the expected Euler characteristic of excursion sets of 2 ; F and t elds. Adv. Appl. Probab. 26, 13–42. Worsley, K.J., 1999. Testing for signals with unknown location and scale in a 2 random eld, with an application to f MRI. Adv. Appl. Probab., in press (subject to revision).
A test for a conjunction ( K.J. Worsley a;∗ , K.J. Fristonb a Department
b Wellcome
of Mathematics and Statistics, McGill University, Montreal, Canada H3A 2K6 Department of Cognitive Neurology, Institute of Neurology, Queen Square, London WC1N 3BG, UK Received March 1999
Abstract A conjunction is de ned in the brain mapping literature as the occurrence of the same event at the same location in two or more independent 3D brain images. The images are smooth isotropic 3D random elds of test statistics, and the event occurs when the image exceeds a xed high threshold. We give a simple approximation to the probability of a conjunction occurring anywhere in a xed region, so that we can test for a local increase in the mean of the images at the same unknown location in all images, a generalization of the split-t test. This is the corollary to a more general result c 2000 Elsevier Science B.V. on the expected Minkowski functionals of the set of points where a conjunction occurs. All rights reserved MSC: 60G60; 62M09 Keywords: Split-t test; Random eld; Euler characteristic; Integral geometry; Stereology
1. Introduction Let Xi (t) be the value of image i at location t ∈ R D ; 16i6n, and let x be a xed threshold. The set of points where a conjunction occurs is C = {t ∈ S: Xi (t)¿x for all 16i6n}: An example is shown in Fig. 1 for n = 6. We are interested in the probability that C is not empty, that is, the probability that all images exceed the threshold at some point inside S, or that the maximum over t ∈ S
( Research supported by the Natural Sciences and Engineering Research Council of Canada, Fonds pour la Formation des Chercheurs et 1’Aide a la Recherche de Quebec, and the Wellcome Trust. ∗
Corresponding author. Tel.: +1-514-398-3842; fax: +1-514-398-3899. E-mail address: [email protected] (K.J. Worsley)
c 2000 Elsevier Science B.V. All rights reserved 0167-7152/00/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 9 9 ) 0 0 1 4 9 - 2
136
K.J. Worsley, K.J. Friston / Statistics & Probability Letters 47 (2000) 135 – 140
Fig. 1. Conjunction of n = 6 f MRI images during a visual task (only one slice of the 3D data is shown). The excursion sets of each Xi (t)¿1:64; i = 1; : : : ; 6 are shown in white on a background of brain anatomy (top). The set of conjunctions C is the intersection of these sets (bottom). The visual cortex at the back of the brain appears in C, but the most interesting feature is the appearance of the lateral geniculate nuclei (LGN) (arrows).
of the minimum over i of Xi (t) exceeds x: P{C 6= ∅} = P max min Xi (t)¿x : t ∈ S 16i6n
(1)
If the images are independent stationary random elds then the expected Lebesgue measure or volume of C is E{|C|}=pn |S|;
(2)
where p = P{Xi (t)¿x}. Our main result is that (2) holds if Lebesgue measure is replaced by a vector of Minkowski functionals, and p is replaced by a matrix of Euler characteristic intensity functions for the random eld. This gives (2) as a special case, and other interesting quantities such as the expected surface area of C, which comes from the (D − 1)-dimensional Minkowski functional. But the component of most interest to us is the zero-dimensional Minkowski functional, or Euler characteristic (EC). For high thresholds, the expected EC of C is a very accurate approximation to probability (1) that we seek (Adler, 1999). We apply this result to some real data in brain mapping in Section 5. This also allows us to set the level of the split-t test. Shaywitz et al. (1995) used this test to determine whether the functional organization of the brain for language di ered according to sex. Thirty eight independent
K.J. Worsley, K.J. Friston / Statistics & Probability Letters 47 (2000) 135 – 140
137
f MRI images were randomly divided into n = 2 groups, and an image Xi (t) of t-statistics was calculated for each group i =1; 2. The split-t test rejects if the null hypothesis is rejected for both groups at the same point t, that is, if X1 (t)¿x and X2 (t)¿x for some threshold x, taken as the upper level 5% point of the t-distribution with 17 degrees of freedom. The resulting image of conjunctions C appears on the cover of Nature that contains Shaywitz et al. (1995). 2. Integral geometry and stereology In this section we shall state some results from integral geometry and stereology that will be used to prove our main result (see, for example, Santalo, 1976). Let i (A) be the ith Minkowski functional of a set A ⊂ R D , scaled so that it is invariant under embedding of A into any higher-dimensional Euclidean space. If A has a twice di erentiable boundary @A, then it can be de ned as follows. Let si = 2i=2 = (i=2) be the surface area of a unit (i − 1)-sphere in R i . For M an m × m matrix let detr j (M ) denote the sum of the determinant of all j × j principal minors of M , so that detr m (M ) = det (M ); detr 1 (M ) = tr(M ) and we de ne detr 0 (M ) = 1. Let Q be the (D − 1) × (D − 1) curvature matrix of @A. Then for 06i6D, Z 1 i (A) = detr D−1−i (Q) dt sD−i @A and de ne D (A) = |A|. Note that 0 (A) is the EC of A by the Gauss–Bonnet Theorem, and D−1 (A) is half the surface area of A. For example, the Minkowski functionals of a ball A of radius r in R 3 are 0 (A) = 1;
1 (A) = 4r;
2 (A) = 2r 2 ;
We shall use the result that any set functional
3 (A) = (4=3)r 3 :
(3)
(A) that obeys the additivity rule
(A ∪ B) = (A) + (B) − (A ∩ B)
(4)
is a linear combination of the Minkowski functionals. Let A; B ⊂ R D , then the Kinematic Fundamental Formula of integral geometry relates the integrated EC of the intersection of A and B to their Minkowski functionals: Z D X i (A)D−i (B) 0 (A ∩ B) = s2 : : : sD ; (5) ciD i=0 where the integral is over all rotations and translations of A, keeping B xed, and ciD =
(1=2) ((D + 1)=2) : ((i + 1)=2) ((D − i + 1)=2)
3. Random elds If X (t); t ∈ R D , is an isotropic random eld with excursion set A = {t: X (t)¿x} then E{0 (A ∩ S)} =
D X
i i (S)
(6)
i=0
for some constant i . This follows from the fact that (S) = E{0 (S ∩ A)} obeys the additivity rule (4), since 0 does, so it must be a linear combination of the Minkowski functionals i (S). The coecients i , called Euler characteristic (EC) intensities in R i , can be evaluated for a variety of random elds (Adler, 1981; Worsley, 1994, 1999; Siegmund and Worsley, 1995; Cao and Worsley, 1999a,b). For example, for a
138
K.J. Worsley, K.J. Friston / Statistics & Probability Letters 47 (2000) 135 – 140
Gaussian random eld with E{X (t)} = 0; Var{X (t)} = 0; Var{@X (t)=@t} = I , where I is the D×D identity matrix, then 0 = P{X (t)¿x} and for i ¿ 0 2
i = i=2 (2)−(i+1)=2 Hei−1 (x)e−x =2 ;
(7)
where Hej (x) is the Hermite polynomial of degree j in x. We now extend (6) to higher Minkowski functionals. Lemma 1. E{i (A ∩ S)} =
D X
cij j−i j (S):
j=i
Proof. Clearly,
(S) = E{i (A ∩ S)} obeys the additivity rule (4), since i does, so we can write
E{i (A ∩ S)} =
D X
aij j (S)
(8)
j=0
for some constants aij that do not depend on S. To evaluate these constants, replace S in (8) by Ek , a bounded convex set in a k-plane, k6D. Since the ith Minkowski functional of a set in R k is zero for i ¿ k, then aik is zero for i ¿ k. For i6k, D
E{i (A ∩ Ek )} X j (Ek ) = → aik aij k (Ek ) k (Ek ) j=i
(9)
as Ek → R k . We can thus interpet aik as the density of the ith Minkowski functional of the excursion set of X in R k . To evaluate aik , apply the Kinematic Fundamental Formula (5) to A ∩ ED and S: Z 0 ((A ∩ ED ) ∩ S) = s2 : : : sD
D X i (A ∩ ED )D−i (S) i=0
ciD
:
Dividing both sides by s2 : : : sD D (ED ) and taking limits, we get E{0 (A ∩ S)} =
D X i=0
lim
ED →R
D
E{i (A ∩ ED )} D−i (S) : D (ED ) ciD
Comparing this with (6) we get lim
ED →R
D
E{i (A ∩ ED )} = ciD D−i D (ED )
and combining this with (9), we get for i6j, aij = cij j−i : Substituting into (8) completes the proof.
K.J. Worsley, K.J. Friston / Statistics & Probability Letters 47 (2000) 135 – 140
139
4. Geometry of the set of conjunctions Theorem 2. Let bi = ((i + 1)=2)= (1=2) and de ne the upper triangular Toeplitz matrix Rk and the vector (B) by 0k =b0 1k =b1 · · · 0 (B)b0 Dk =bD 0 (B)b 0k =b0 · · · (D−1)k =bD−1 1 1 ; ; (B) = Rk = . . . . .. .. .. .. . .. 0 0 ··· 0k =b0 D (B)bD where ik is the EC intensity of Xk (t) in R i ; 16k6n, and B ⊂ R D . Then ! n Y Ri (S): E{(C)} = i=1
Proof. The proof follows by induction on n. From the lemma, we see that it is clearly true for n = 1. Let Ak be the excursion set for Xk (t), so that C = A1 ∩ · · · ∩ An ∩ S. If the result is true for n = k then by rst conditioning on Ak+1 and replacing S by Ak+1 ∩ S we get ! ! k k Y Y E{(A1 ∩ · · · ∩ Ak ∩ (Ak+1 ∩ S)} = Ri E{(Ak+1 ∩ S)} = Ri Rk+1 (S) i=1
i=1
by the result for n = 1. This completes the proof. Comparing this result with (2) we see that it has the same form, with volume replaced by the vector of weighted Minkowski functionals, and probability replaced by the matrix of weighted EC intensities. The last element is the same as in (2), and the rst element is the expected EC of the set of conjunctions that we shall use as an approximation to the probability of a conjunction anywhere in S: ! n Y (10) Ri (S) P{C 6= 0} = P max min Xi (t)¿x ≈ E{0 (C)} = (1; 0; : : : ; 0) t ∈ S 16i6n
i=1
for high thresholds x. 5. Application We shall apply the result to some D = 3 dimensional functional magnetic resonance imaging (f MRI) data fully described in Friston et al. (1999). The purpose of the experiment was to determine those regions of the brain that were consistently stimulated by all subjects while viewing a pattern of radially moving dots. To do this, subjects were presented with a pattern of moving dots, followed by a pattern of stationary dots, and this was repeated 10 times, during which a total of 120 3D f MRI images were obtained at the rate of one every 3.22 s. For each subject i and at every point t ∈ R 3 , a test statistic Xi (t) was calculated for comparing the f MRI response between the moving dots and the stationary dots. Under the null hypothesis of no di erence, Xi (t) was modeled as an isotropic Gaussian random eld with zero mean, unit variance and = 4:68 cm−2 . A threshold of x = 1:64, corresponding to an uncorrected level 5% test, was chosen, and the excursion sets for each subject are shown in Fig. 1, together with their intersection, which forms the set of conjunctions C. The search region S is the whole brain area that was scanned, which was an approximate spherical region with a volume of |S| = 1226 cm3 . Finally, the approximate probability of a conjunction, calculated from (3), (7)
140
K.J. Worsley, K.J. Friston / Statistics & Probability Letters 47 (2000) 135 – 140
and (10), is 0.0126. We can thus conclude, at the 1.26% level, that conjunctions have occured in the visual cortex, and more interesting, the lateral geniculate nuclei (see Fig. 1). References Adler, R.J., 1981. The Geometry of Random Fields. Wiley, New York. Adler, R.J., 1999. On excursion sets, tube formulae, and maxima of random elds. Ann. Appl. Probab., in press. Cao, J., Worsley, K.J., 1999a. The detection of local shape changes via the geometry of Hotelling’s T 2 elds. Ann. Statist., in press. Cao, J., Worsley, K.J., 1999b. The geometry of correlation elds with an application to functional connectivity of the brain. Ann. Appl. Probab., in press. Friston, K.J., Holmes, A.P., Price, C.J., Buchel, C., Worsley, K.J., 1999. Multi-subject f MRI studies and conjunction analyses. Neuroimage 10, 385–396. Santalo, L.A., 1976. Integral geometry and geometric probability. In: Rota, G.-C. (Ed.), Encyclopedia of Mathematics and its Applications, Vol. 1. Addison-Wesley, Reading, MA. Shaywitz, B.A., Shaywitz, S.E., Pugh, K.R., 1995. Sex di erences in the functional organization of the brain for language. Nature 373, 607–609. Siegmund, D.O., Worsley, K.J., 1995. Testing for a signal with unknown location and scale in a stationary Gaussian random eld. Ann. Statist. 23, 608–639. Worsley, K.J., 1994. Local maxima and the expected Euler characteristic of excursion sets of 2 ; F and t elds. Adv. Appl. Probab. 26, 13–42. Worsley, K.J., 1999. Testing for signals with unknown location and scale in a 2 random eld, with an application to f MRI. Adv. Appl. Probab., in press (subject to revision).